Question

Out of 400 people sampled, 96 received flu vaccinations this year. Based on this, construct a 99% confidence interval for the true population proportion of people who received flu vaccinations this year. Give your answers as decimals, to three places .......

Answer 1

The 99% confidence interval for the true population proportion of people

who received flu vaccinations this year is approximately
`\( \left(0.202, 0.298\right) \).`

Step-by-step explanation:

The confidence interval is calculated using the formula:

Confidence Interval =
`\hat{p} \pm Z * \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]`

-
`\(\hat{p}\)` is the sample proportion (96/400),

- (Z) is the Z-score corresponding to the desired confidence level (for 99%,
`\(Z \approx 2.576\)`for a two-tailed test),

- n is the sample size (400).

Substituting these values into the formula, we get the confidence interval.

Now, interpreting the interval: We are 99% confident that the true proportion of people who received flu vaccinations this year lies between 20.2% and 29.8%. or 0.202 and 0.298

This means that if we were to take many random samples and construct a confidence interval for each sample, we would expect about 99% of those intervals to contain the true population proportion.

Remember that confidence intervals give us a range of plausible values for the parameter we are estimating, providing a measure of the precision of our estimate. In this case, we are estimating the proportion of people who received flu vaccinations this year in the entire population.